Introducing the mean field approximation to CDFT/MMpol method: Statistically converged equilibrium and nonequilibrium free energy calculation for electron transfer reactions in condensed phases.

A new theoretical method to study electron transfer reactions in condensed phases is proposed by introducing the mean-field approximation into the constrained density functional theory/molecular mechanical method with a polarizable force field (CDFT/MMpol). The method enables us to efficiently calculate the statistically converged equilibrium and nonequilibrium free energies for diabatic states in an electron transfer reaction by virtue of the mean field approximation that drastically reduces the number of CDFT calculations. We apply the method to the system of a formanilide-anthraquinone dyad in dimethylsulfoxide, in which charge recombination and cis-trans isomerization reactions can take place, previously studied by the CDFT/MMpol method. Quantitative agreement of the driving force and the reorganization energy between our results and those from the CDFT/MMpol calculation and the experimental estimates supports the utility of our method. The calculated nonequilibrium free energy is analyzed by its decomposition into several contributions such as those from the averaged solute-solvent electrostatic interactions and the explicit solvent electronic polarization. The former contribution is qualitatively well described by a model composed of a coarse-grained dyad in a solution in the linear response regime. The latter contribution reduces the reorganization energy by more than 10 kcal/mol.